Download The Quadratic Equation

The Quadratic Equation
• A quadratic equation is an equation of the form:
ax 2  bx  c  0, a  0.
• One way to solve a quadratic equation is by factoring.
• Example. Solve 3x2 + 5x – 2 = 0 by factoring.
(3x  1)( x  2)  0
3x  1  0 or x  2  0
1
x
or
x  2.
3
• Note that we are using the following property which holds for
complex numbers a and b.
If ab = 0, then a = 0 or b = 0.
Completing the Square
• Another method for solving a quadratic equation involves
completing the square, which we show by solving 2x2–10x +1 = 0.
2 x 2  10 x  1
2( x 2  5 x)  1
1
2
x  5x  
2
25
 5
Add 
to both sides.
 
4
 2 
25
1 25
2
x  5x 
 
4
2 4
2
2
5
23

x  
2
4

5
23
x 
2
2
The Quadratic Formula
• By completing the square on the general quadratic, we can
obtain the quadratic formula, which is displayed next.
• The quadratic equation
has solutions
ax 2  bx  c  0, a  0.
 b  b 2  4ac
x
.
2a
• Example. Solve 2x2–10x +1 = 0. Here, a = 2, b = –10, c = 1.
The solution obtained from the quadratic formula is:
10  100  (4)( 2) 5  23
x

,
4
2
which agrees with the result we got by completing the square.
The Discriminant
• The discriminant is the expression b2– 4ac found under the
radical in the quadratic formula.
• If b2– 4ac is negative, we have the square root of a negative
number, and the roots are complex conjugate pairs.
• If b2– 4ac is positive, we have the square root of a positive
number, and the roots are two different real numbers.
• If b2– 4ac = 0, then x = – b/2a. We say that the equation has a
double root or repeated root in this case.
• Example. What does the discriminant tell you about the
equation 4x2 –20x + 25 = 0?
Radical Equations
• If an equation involving x (and no higher powers of x) and a
single radical, we proceed as follows to solve the equation.
• First isolate the radical on one side of the equation.
• Second, square both sides of the resulting equation to obtain a
quadratic equation.
• Third, solve the quadratic, but be sure to check your answers
in the original equation since squaring both sides may have
introduced extraneous solutions.
• Example. Solve x 
x  2  4.
Quadratic Equations and Word Problems
• We now consider a group of applied problems that lead to
quadratic equations.
• Problem. The larger of two positive numbers exceeds the
smaller by 2. If the sum of the squares of the numbers is 74,
find the numbers.
Solution. Let x = the larger number, x – 2 = the smaller number.
x 2  ( x  2) 2  74
2 x 2  4 x  70  0
x 2  2 x  35  0
( x  5)( x  7)  0
x  7, reject x  5.
More Quadratic Equations and Word Problems
• Problem. Working together, computers A and B can complete
a data-processing job in 2 hours. Computer A working alone
can do the job in 3 hours less than computer B working alone.
How long does it take each computer to do the job by itself?
Solution. Let x = time for B alone, x – 3 = time for A alone.
Then the rate for B is 1/x and the rate for A is 1/(x – 3).
2
2
  1 whole job
x 3 x
2
 2
x( x-3) 
   x( x-3)
x 3 x
x2  7x  6  0
( x  6)( x  1)  0
x  6 or x  1
x  3  3, x  3  2.
The Quadratic Equation; We discussed
• The definition of a quadratic equation
• Solving by factoring
• Completing the square
• The quadratic formula
• The discriminant and its use in predicting the nature of the roots
• Radical equations and extraneous solutions
• Word problems for quadratic equations